3.276 \(\int \frac {\sqrt {c+d x^2}}{\sqrt {-a+b x^2}} \, dx\)

Optimal. Leaf size=88 \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {b x^2-a} \sqrt {\frac {d x^2}{c}+1}} \]

[Out]

EllipticE(x*b^(1/2)/a^(1/2),(-a*d/b/c)^(1/2))*a^(1/2)*(1-b*x^2/a)^(1/2)*(d*x^2+c)^(1/2)/b^(1/2)/(b*x^2-a)^(1/2
)/(1+d*x^2/c)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {427, 426, 424} \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {b x^2-a} \sqrt {\frac {d x^2}{c}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x^2]/Sqrt[-a + b*x^2],x]

[Out]

(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*
Sqrt[-a + b*x^2]*Sqrt[1 + (d*x^2)/c])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 427

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x^2}}{\sqrt {-a+b x^2}} \, dx &=\frac {\sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x^2}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{\sqrt {-a+b x^2}}\\ &=\frac {\left (\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}\right ) \int \frac {\sqrt {1+\frac {d x^2}{c}}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{\sqrt {-a+b x^2} \sqrt {1+\frac {d x^2}{c}}}\\ &=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {-a+b x^2} \sqrt {1+\frac {d x^2}{c}}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 88, normalized size = 1.00 \[ \frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {c+d x^2} E\left (\sin ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} \sqrt {b x^2-a} \sqrt {\frac {c+d x^2}{c}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x^2]/Sqrt[-a + b*x^2],x]

[Out]

(Sqrt[(a - b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], -((a*d)/(b*c))])/(Sqrt[b/a]*Sqrt[-a + b*x^
2]*Sqrt[(c + d*x^2)/c])

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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} - a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(1/2)/(b*x^2-a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^2 + c)/sqrt(b*x^2 - a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} - a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(1/2)/(b*x^2-a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^2 + c)/sqrt(b*x^2 - a), x)

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maple [B]  time = 0.02, size = 168, normalized size = 1.91 \[ \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}-a}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {-\frac {b \,x^{2}-a}{a}}\, \left (-a d \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )+a d \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )+b c \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )\right )}{\left (b d \,x^{4}-a d \,x^{2}+b c \,x^{2}-a c \right ) \sqrt {-\frac {d}{c}}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^(1/2)/(b*x^2-a)^(1/2),x)

[Out]

(d*x^2+c)^(1/2)*(b*x^2-a)^(1/2)*((d*x^2+c)/c)^(1/2)*(-(b*x^2-a)/a)^(1/2)*(a*d*EllipticF((-1/c*d)^(1/2)*x,(-1/a
*b*c/d)^(1/2))+c*EllipticF((-1/c*d)^(1/2)*x,(-1/a*b*c/d)^(1/2))*b-a*d*EllipticE((-1/c*d)^(1/2)*x,(-1/a*b*c/d)^
(1/2)))/(b*d*x^4-a*d*x^2+b*c*x^2-a*c)/(-1/c*d)^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{2} + c}}{\sqrt {b x^{2} - a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(1/2)/(b*x^2-a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^2 + c)/sqrt(b*x^2 - a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {d\,x^2+c}}{\sqrt {b\,x^2-a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^(1/2)/(b*x^2 - a)^(1/2),x)

[Out]

int((c + d*x^2)^(1/2)/(b*x^2 - a)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c + d x^{2}}}{\sqrt {- a + b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**(1/2)/(b*x**2-a)**(1/2),x)

[Out]

Integral(sqrt(c + d*x**2)/sqrt(-a + b*x**2), x)

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